The Laguerre process and generalized Hartman--Watson law

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In this paper, we study complex Wishart processes or the so-called Laguerre processes (Xₜ)ₜ≥₀. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman--Watson law as well as the law of T₀:={t,(Xₜ)=0} when the size of the matrix is 2.